14-Arbitrage Page 423 Wednesday, February 4, 2004 1:08 PM
------------- -------------------- -------------------- ------------- -------------------- --------------------Arbitrage Pricing: Finite-State Models 423d 1
i
() 1
R 01 ,p 1 π 2 () 1
πA 10 ,- R 02 ,
p 2 π 2 () 2
πA 10 ,+ -R 02 ,
d 2
i
() 3
R 01 ,p 3 π 2 () 3
πA 10 ,- R 02 ,
p 4 π 2 () 4
πA 10 ,= + + -R 02 ,d 2 1 2 3 4
i
() d 2
i
() d 2
i
() d 2
i
+ q ()
1 --------------+ q 2 --------------+ q 3 --------------+ q 4 --------------
R 02 , R 02 , R 02 , R 02 ,d 1 1 2 3 4
i
() d 1
i
() d 1
i
() d 1
i
= q ()
1 --------------+ q 2 --------------+ q 3 --------------+ q 4 --------------
R 01 , R 01 , R 01 , R 01 ,d 2 1 2 3 4
i
() d 2
i
() d 2
i
() d 2
i
+ q ()
1 --------------+ q 2 --------------+ q 3 --------------+ q 4 --------------
R 02 , R 02 , R 02 , R 02 ,PATH DEPENDENCE AND MARKOV MODELS
The value of a derivative instrument might depend on the path of its past
values. Consider a lookback option on a stock—that is, a derivative
instrument on a stock whose payoff at time t is the maximum difference
between the price of the stock and a given value K at any moment prior to
t. Call Vt the payoff of the lookback option at time t. We can then write:Vt = max (Sk – K)+
0 kt< ≤The notation (Sk – K)
+
means Sk – K if the difference is positive, 0 oth-
erwise, that is, (Sk – K)
+
= max(Sk – K, 0 ). Because its value depends
on the entire path taken by the underlying stock, a lookback option is a
path-dependent security.
An adapted process Xt is said to be a Markov process if its condi-
tional distribution at time t depends only on the value of the process at
time t – 1 and not on the value of the process at dates t – 2, t – 3, .... The
Markov property can be formally stated as follows:PX( tXt – 1 )= PX( tXt – 1 , Xt – 2 , ...,X 0 )THE BINOMIAL MODEL
Let’s now introduce the simple but important multiperiod finite-state
model known as the binomial model. The binomial model is important